CircleMedium
Question
Let $y = x$ be the equation of a chord of the circle $\mathbf{C}_{1}$ (in the closed half-plane $x \geq 0$ ) of diameter 10 passing through the origin. Let $C_{2}$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $C_{2}$, which passes through the point $(2,3)$ and is farthest from the center of $C_{2}$, is $x + ay + b = 0$, then $a - b$ is equal to :
Options
A.10
B.-6
C.-2
D.6
Solution
Equation of circle $C_{2}$ is
$$x^{2} + y^{2} - 5x - 5y = 0 $$its centre is $\left( \frac{5}{2},\frac{5}{2} \right)$
$$m_{AB} = - 1 $$∴ Slope of required chord $= 1$
∴ equation of required chord is $x - y + 1 = 0$
$${\therefore a = - 1,\text{ }b = 2 }{\therefore a - b = - 2}$$
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